CGAL 5.1.3  Algebraic Foundations

A model of EuclideanRing
represents an euclidean ring (or Euclidean domain). It is an UniqueFactorizationDomain
that affords a suitable notion of minimality of remainders such that given \( x\) and \( y \neq 0\) we obtain an (almost) unique solution to \( x = qy + r \) by demanding that a solution \( (q,r)\) is chosen to minimize \( r\). In particular, \( r\) is chosen to be \( 0\) if possible.
Moreover, CGAL::Algebraic_structure_traits< EuclideanRing >
is a model of AlgebraicStructureTraits
providing:
CGAL::Algebraic_structure_traits< EuclideanRing >::Algebraic_category
derived from CGAL::Unique_factorization_domain_tag
CGAL::Algebraic_structure_traits< EuclideanRing >::Mod
which is a model of AlgebraicStructureTraits_::Mod
CGAL::Algebraic_structure_traits< EuclideanRing >::Div
which is a model of AlgebraicStructureTraits_::Div
CGAL::Algebraic_structure_traits< EuclideanRing >::Div_mod
which is a model of AlgebraicStructureTraits_::DivMod
Remarks
The most prominent example of a Euclidean ring are the integers. Whenever both \( x\) and \( y\) are positive, then it is conventional to choose the smallest positive remainder \( r\).